U.S. patent application number 16/068405 was filed with the patent office on 2019-01-17 for techniques for manipulation of two-qubit quantum states and related systems and methods.
This patent application is currently assigned to Yale University. The applicant listed for this patent is YALE UNIVERSITY. Invention is credited to Michel DEVORET, Luigi FRUNZIO, Yvonne GAO, Robert J. SCHOELKOPF, III, Chen WANG.
Application Number | 20190020346 16/068405 |
Document ID | / |
Family ID | 59311688 |
Filed Date | 2019-01-17 |
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United States Patent
Application |
20190020346 |
Kind Code |
A1 |
WANG; Chen ; et al. |
January 17, 2019 |
TECHNIQUES FOR MANIPULATION OF TWO-QUBIT QUANTUM STATES AND RELATED
SYSTEMS AND METHODS
Abstract
According to some aspects, a method is provided of operating a
system that includes a multi-level quantum system dispersively
coupled to a first quantum mechanical oscillator and dispersively
coupled to a second quantum mechanical oscillator, the method
comprising applying a first drive waveform to the multi-level
quantum system, applying one or more second drive waveforms to the
first quantum mechanical oscillator, and applying one or more third
drive waveforms to the second quantum mechanical oscillator.
Inventors: |
WANG; Chen; (New Haven,
CT) ; GAO; Yvonne; (New Haven, CT) ; FRUNZIO;
Luigi; (North Haven, CT) ; DEVORET; Michel;
(North Haven, CT) ; SCHOELKOPF, III; Robert J.;
(Madison, CT) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
YALE UNIVERSITY |
New haven |
CT |
US |
|
|
Assignee: |
Yale University
New haven
CT
|
Family ID: |
59311688 |
Appl. No.: |
16/068405 |
Filed: |
January 13, 2017 |
PCT Filed: |
January 13, 2017 |
PCT NO: |
PCT/US2017/013426 |
371 Date: |
July 6, 2018 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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62335591 |
May 12, 2016 |
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62279624 |
Jan 15, 2016 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
H03B 5/1847 20130101;
B82Y 10/00 20130101; G06N 10/00 20190101; H03K 19/195 20130101 |
International
Class: |
H03K 19/195 20060101
H03K019/195; H03B 5/18 20060101 H03B005/18; G06N 99/00 20060101
G06N099/00 |
Goverment Interests
GOVERNMENT FUNDING
[0002] This invention was made with government support under
W911NF-14-1-0011 awarded by United States Army Research Office. The
government has certain rights in the invention.
Claims
1. A method of operating a system that includes a multi-level
quantum system dispersively coupled to a first quantum mechanical
oscillator and dispersively coupled to a second quantum mechanical
oscillator, the method comprising: applying a first drive waveform
to the multi-level quantum system; applying one or more second
drive waveforms to the first quantum mechanical oscillator; and
applying one or more third drive waveforms to the second quantum
mechanical oscillator.
2. The method of claim 1, wherein: the first quantum mechanical
oscillator implements a first logical qubit, the second quantum
mechanical oscillator implements a second logical qubit, and the
first drive waveform, one or more second drive waveforms and one or
more third drive waveforms are together configured to perform a
quantum logic gate between the first logical qubit and the second
logical qubit.
3. The method of claim 1, wherein the multi-level quantum system is
a nonlinear quantum system.
4. The method of claim 1, wherein: the first drive waveform is
configured to produce a superposition of states of the multi-level
quantum system, the one or more second drive waveforms are
configured to coherently add or remove energy to or from the first
quantum mechanical oscillator conditional on a state of the
multi-level quantum system, and the one or more third drive
waveforms are configured to coherently add or remove energy to or
from the second quantum mechanical oscillator conditional on the
state of the multi-level quantum system.
5. The method of claim 4, wherein the state of the multi-level
quantum system is a superposition of a ground state and a first
excited state.
6. The method of claim 5, wherein the one or more second drive
waveforms and one or more third drive waveforms are configured to
coherently add or remove energy conditional on whether the
multi-level quantum system is in the ground state or in the first
excited state.
7. The method of claim 5, wherein the one or more second drive
waveforms consist of a single drive waveform with a bandwidth
smaller than a dispersive frequency shift of the first quantum
mechanical oscillator associated with a transition between the
ground state and the first excited state of the multi-level quantum
system.
8. The method of claim 1, wherein applying the one or more second
drive waveforms comprises: applying an initial drive waveform to
the first quantum mechanical oscillator that coherently adds or
removes energy to or from the first quantum mechanical oscillator;
waiting for a predetermined time subsequent to application of the
initial drive waveform; and applying a subsequent drive waveform to
the first quantum mechanical oscillator that coherently adds or
removes energy to or from the first quantum mechanical
oscillator.
9. The method of claim 1, wherein prior to application of the first
drive waveform, the multi-level quantum system, the first quantum
mechanical oscillator, and the second quantum mechanical oscillator
are in respective ground states.
10. The method of claim 4, further comprising: applying a fourth
drive waveform to the multi-level quantum system, the fourth drive
waveform configured to change the state of the multi-level system
conditional on a state of the first quantum mechanical oscillator
and a state of the second quantum mechanical oscillator.
11. The method of claim 1, further comprising measuring joint
parity of the first quantum mechanical oscillator and second
quantum mechanical oscillator.
12. The method of claim 11, wherein measuring joint parity of the
first quantum mechanical oscillator and the second quantum
mechanical oscillator comprises: applying a fifth drive waveform to
the multi-level quantum system; waiting for a first predetermined
time subsequent to application of the fifth drive waveform;
applying a sixth drive waveform to the multi-level quantum system;
waiting for a second predetermined time subsequent to application
of the sixth drive waveform; and applying a seventh drive waveform
to the multi-level quantum system.
13. The method of claim 1, further comprising measuring a state of
the multi-level quantum system via a readout resonator coupled to
the multi-level quantum system.
14. The method of claim 13, wherein the readout resonator is
capacitively coupled to the multi-level quantum system and wherein
the readout resonator is further coupled to a transmission
line.
15. The method of claim 13, wherein measuring the state of the
multi-level quantum system via the readout resonator comprises
measuring an amplitude and a phase of a signal output from the
readout resonator.
16. The method of claim 1, wherein
.chi..sub.B.sup.ge>.chi..sub.A.sup.ge, where .chi..sub.A.sup.ge
is a dispersive frequency shift of the first quantum mechanical
oscillator associated with a transition between a ground state of
the multi-level quantum system and a first excited state of the
multi-level quantum system, and where .chi..sub.B.sup.ge is a
dispersive frequency shift of the second quantum mechanical
oscillator associated with the transition between the ground state
of the multi-level quantum system and the first excited state of
the multi-level quantum system.
17. The method of claim 1, wherein the multi-level quantum system
is a superconducting transmon.
18. The method of claim 1, wherein the first quantum mechanical
oscillator and the second quantum mechanical oscillator are
resonator cavities.
19. A circuit quantum electrodynamics system, comprising: a
multi-level quantum system; a first quantum mechanical oscillator
dispersively coupled to the multi-level quantum system; a second
quantum mechanical oscillator dispersively coupled to the
multi-level quantum system; and at least one electromagnetic
radiation source configured to apply independent electromagnetic
pulses to the multi-level quantum system, to the first quantum
mechanical oscillator, and to the second quantum mechanical
oscillator.
20. The system of claim 19, wherein the at least one
electromagnetic radiation source is configured to: apply a first
drive waveform to the multi-level quantum system, the first drive
waveform configured to produce a superposition of states of the
multi-level quantum system; apply one or more second drive
waveforms to the first quantum mechanical oscillator, the one or
more second drive waveforms configured to coherently add or remove
energy to or from the first quantum mechanical oscillator
conditional on a state of the multi-level quantum system; and apply
one or more third drive waveforms to the second quantum mechanical
oscillator, the one or more third drive waveforms configured to
coherently add or remove energy to or from the second quantum
mechanical oscillator conditional on the state of the multi-level
quantum system.
21. The system of claim 19, wherein the multi-level quantum system
is a nonlinear quantum system.
22. The system of claim 21, wherein the nonlinear quantum system
comprises: a Josephson junction comprising a first superconducting
portion, a second superconducting portion, and an insulating
portion, wherein the first superconducting portion and the second
superconducting portion are physically separated by the insulating
portion; a first antenna electrically connected to the first
superconducting portion; a second antenna electrically connected to
the first superconducting portion; and a third antenna electrically
connected to the second superconducting portion.
23. The system of claim 22, wherein the first quantum mechanical
oscillator is dispersively coupled to the nonlinear quantum system
via the first antenna, and wherein the second quantum mechanical
oscillator is dispersively coupled to the nonlinear quantum system
via the second antenna.
24. The system of claim 19, wherein the multi-level quantum system
is a superconducting transmon.
25. The system of claim 19, wherein the first quantum mechanical
oscillator and the second quantum mechanical oscillator are
resonator cavities.
26. The system of claim 19, further comprising a stripline readout
resonator capacitively coupled to the multi-level quantum
system.
27. The system of claim 20, wherein at least one of the first drive
waveform, the one or more second drive waveforms, and the one or
more third drive waveforms are produced by a field programmable
gate array (FPGA).
28. A nonlinear quantum device comprising: a Josephson junction
comprising a first superconducting portion, a second
superconducting portion, and an insulating portion, wherein the
first superconducting portion and the second superconducting
portion are physically separated by the insulating portion; a first
antenna electrically connected to the first superconducting
portion; a second antenna electrically connected to the first
superconducting portion; and a third antenna electrically connected
to the second superconducting portion.
29. The nonlinear quantum device of claim 28, wherein the first
antenna, the second antenna and the first superconducting portion
intersect at a single location.
30. The nonlinear quantum device of claim 28, further comprising a
metallic strip capacitively coupled to the third antenna.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] The present application claims the benefit under 35 U.S.C.
.sctn. 119(e) of U.S. Provisional Patent Application No.
62/279,624, filed Jan. 15, 2016, titled "Methods and Apparatus for
Manipulation of Multi-Cavity Quantum States," and U.S. Provisional
Patent Application No. 62/335,591, filed May 12, 2016, titled
"Methods and Apparatus for Manipulation of Multi-Cavity Quantum
States," each of which is hereby incorporated by reference in its
entirety.
BACKGROUND
[0003] The ability to prepare and control the quantum state of a
quantum system is important for quantum information processing.
Just as a classical computer memory should have the ability to
initialize bits and implement gates to change the state of a bit
from zero to one and vice versa, a quantum computer should be able
to initialize the state of the quantum system used to store quantum
information and the quantum system should be able to be controlled
to implement logical gates that change the quantum state of the
quantum system.
[0004] Quantum information may be stored in any of a variety of
quantum mechanical systems. Conventionally, quantum information may
be stored using quantum bits, referred to as "qubits," which are
typically two-state quantum mechanical systems. However, many-state
quantum systems, such as quantum mechanical oscillators, may also
be used to store quantum information.
SUMMARY
[0005] According to some aspects, a method is provided of operating
a system that includes a multi-level quantum system dispersively
coupled to a first quantum mechanical oscillator and dispersively
coupled to a second quantum mechanical oscillator, the method
comprising applying a first drive waveform to the multi-level
quantum system, applying one or more second drive waveforms to the
first quantum mechanical oscillator, and applying one or more third
drive waveforms to the second quantum mechanical oscillator.
[0006] According to some embodiments, the first quantum mechanical
oscillator implements a first logical qubit, the second quantum
mechanical oscillator implements a second logical qubit, and the
first drive waveform, one or more second drive waveforms and one or
more third drive waveforms are together configured to perform a
quantum logic gate between the first logical qubit and the second
logical qubit.
[0007] According to some embodiments, the multi-level quantum
system is a nonlinear quantum system.
[0008] According to some embodiments, the first drive waveform is
configured to produce a superposition of states of the multi-level
quantum system, the one or more second drive waveforms are
configured to coherently add or remove energy to or from the first
quantum mechanical oscillator conditional on a state of the
multi-level quantum system, and the one or more third drive
waveforms are configured to coherently add or remove energy to or
from the second quantum mechanical oscillator conditional on the
state of the multi-level quantum system.
[0009] According to some embodiments, the state of the multi-level
quantum system is a superposition of a ground state and a first
excited state.
[0010] According to some embodiments, the one or more second drive
waveforms and one or more third drive waveforms are configured to
coherently add or remove energy conditional on whether the
multi-level quantum system is in the ground state or in the first
excited state.
[0011] According to some embodiments, the one or more second drive
waveforms consist of a single drive waveform with a bandwidth
smaller than a dispersive frequency shift of the first quantum
mechanical oscillator associated with a transition between the
ground state and first excited state of the multi-level quantum
system.
[0012] According to some embodiments, applying the one or more
second drive waveforms comprises applying an initial drive waveform
to the first quantum mechanical oscillator that coherently adds or
removes energy to or from the first quantum mechanical oscillator,
waiting for a predetermined time subsequent to application of the
initial drive waveform, and applying a subsequent drive waveform to
the first quantum mechanical oscillator that coherently adds or
removes energy to or from the first quantum mechanical
oscillator.
[0013] According to some embodiments, prior to application of the
first drive waveform, the multi-level quantum system, first quantum
mechanical oscillator, and second quantum mechanical oscillator are
in respective ground states.
[0014] According to some embodiments, the method further comprises
applying a fourth drive waveform to the multi-level quantum system,
the fourth drive waveform configured to change the state of the
multi-level system conditional on states of the first and second
quantum mechanical oscillators.
[0015] According to some embodiments, the method further comprises
measuring joint parity of the first quantum mechanical oscillator
and second quantum mechanical oscillator.
[0016] According to some embodiments, measuring joint parity of the
first quantum mechanical oscillator and second quantum mechanical
oscillator comprises applying a fifth drive waveform to the
multi-level quantum system, waiting for a first predetermined time
subsequent to application of the fifth drive waveform, applying a
sixth drive waveform to the multi-level quantum system, waiting for
a second predetermined time subsequent to application of the sixth
drive waveform, and applying a seventh drive waveform to the
multi-level quantum system.
[0017] According to some embodiments, the method further comprises
measuring a state of the multi-level quantum system via a readout
resonator coupled to the multi-level quantum system.
[0018] According to some embodiments, the readout resonator is
capacitively coupled to the multi-level quantum system and the
readout resonator is further coupled to a transmission line.
[0019] According to some embodiments, measuring the state of the
multi-level quantum system via the readout resonator comprises
measuring an amplitude and a phase of a signal output from the
readout resonator.
[0020] According to some embodiments, where is a dispersive
frequency shift of the first quantum mechanical oscillator
associated with a transition between a ground state of the
multi-level quantum system and a first excited state of the
multi-level quantum system, and where is a dispersive frequency
shift of the second quantum mechanical oscillator associated with a
transition between the ground state of the multi-level quantum
system and the first excited state of the multi-level quantum
system.
[0021] According to some embodiments, the multi-level quantum
system is a superconducting transmon.
[0022] According to some embodiments, the first quantum mechanical
oscillator and second quantum mechanical oscillator are resonator
cavities.
[0023] According to some aspects, a circuit quantum electrodynamics
system is provided, comprising a multi-level quantum system, a
first quantum mechanical oscillator dispersively coupled to the
multi-level quantum system, a second quantum mechanical oscillator
dispersively coupled to the multi-level quantum system, and at
least one electromagnetic radiation source configured to apply
independent electromagnetic pulses to the multi-level quantum
system, to the first quantum mechanical oscillator, and to the
second quantum mechanical oscillator.
[0024] According to some embodiments, the at least one
electromagnetic radiation source is configured to apply a first
drive waveform to the multi-level quantum system, the first drive
waveform configured to produce a superposition of states of the
multi-level quantum system, apply one or more second drive
waveforms to the first quantum mechanical oscillator, the one or
more second drive waveforms configured to coherently add or remove
energy to or from the first quantum mechanical oscillator
conditional on a state of the multi-level quantum system, and apply
one or more third drive waveforms to the second quantum mechanical
oscillator, the one or more third drive waveforms configured to
coherently add or remove energy to or from the second quantum
mechanical oscillator conditional on the state of the multi-level
quantum system.
[0025] According to some embodiments, the multi-level quantum
system is a nonlinear quantum system.
[0026] According to some embodiments, the multi-level quantum
system is a nonlinear quantum system comprising a Josephson
junction comprising a first superconducting portion, a second
superconducting portion, and an insulating portion, the first
superconducting portion and the second superconducting portion are
physically separated by the insulating portion, a first antenna
electrically connected to the first superconducting portion, a
second antenna electrically connected to the first superconducting
portion, and a third antenna electrically connected to the second
superconducting portion.
[0027] According to some embodiments, the first quantum mechanical
oscillator is dispersively coupled to the nonlinear quantum system
via the first antenna, and the second quantum mechanical oscillator
is dispersively coupled to the nonlinear quantum system via the
second antenna.
[0028] According to some embodiments, the multi-level quantum
system is a superconducting transmon.
[0029] According to some embodiments, the first quantum mechanical
oscillator and second quantum mechanical oscillator are resonator
cavities.
[0030] According to some embodiments, the system further comprises
a stripline readout resonator capacitively coupled to the
multi-level quantum system.
[0031] According to some embodiments, at least one of the first
drive waveform, the one or more second drive waveforms, and the one
or more third drive waveforms are produced by a field programmable
gate array (FPGA).
[0032] According to some aspects, a nonlinear quantum device is
provided comprising a Josephson junction comprising a first
superconducting portion, a second superconducting portion, and an
insulating portion, the first superconducting portion and the
second superconducting portion are physically separated by the
insulating portion, a first antenna electrically connected to the
first superconducting portion, a second antenna electrically
connected to the first superconducting portion, and a third antenna
electrically connected to the second superconducting portion.
[0033] According to some embodiments, the first antenna, the second
antenna and the first superconducting portion intersect at a single
location.
[0034] According to some embodiments, the nonlinear quantum device
further comprises a metallic strip capacitively coupled to the
third antenna.
[0035] The foregoing apparatus and method embodiments may be
implemented with any suitable combination of aspects, features, and
acts described above or in further detail below. These and other
aspects, embodiments, and features of the present teachings can be
more fully understood from the following description in conjunction
with the accompanying drawings.
BRIEF DESCRIPTION OF DRAWINGS
[0036] Various aspects and embodiments will be described with
reference to the following figures. It should be appreciated that
the figures are not necessarily drawn to scale. In the drawings,
each identical or nearly identical component that is illustrated in
various figures is represented by a like numeral. For purposes of
clarity, not every component may be labeled in every drawing.
[0037] FIG. 1 is a block diagram of a circuit quantum
electrodynamics system suitable for practicing aspects of the
present disclosure;
[0038] FIG. 2 illustrates a control sequence for producing cat
states spanning modes of two quantum mechanical oscillators,
according to some embodiments;
[0039] FIG. 3A is a three-dimensional schematic of an illustrative
circuit quantum electrodynamics system comprising two coaxial
resonator cavities and a readout resonator coupled to an ancilla
transmon, according to some embodiments;
[0040] FIG. 3B is a top view of the system shown in FIG. 3A,
according to some embodiments;
[0041] FIG. 4 illustrates an alternative control sequence suitable
for producing cat states spanning modes of two quantum mechanical
oscillators, according to some embodiments;
[0042] FIG. 5 illustrates a control sequence suitable for producing
cat states spanning modes of two quantum mechanical oscillators in
which conditional displacements upon the oscillators are each
realized by two non-conditional displacements, according to some
embodiments;
[0043] FIGS. 6A-6B are illustrative control sequences depicting two
approaches to experimentally measuring the joint parity of two
quantum mechanical oscillators, according to some embodiments;
[0044] FIG. 7 is a circuit diagram of an illustrative experimental
setup for controlling and/or measuring a system of two quantum
mechanical oscillators coupled to an ancilla multi-level quantum
system, according to some embodiments;
[0045] FIG. 8A is a photograph of a machined aluminum package
containing two coaxial stub cavity resonators and a transmon,
suitable for practicing aspects of the present disclosure;
[0046] FIG. 8B is a micrograph image of the transmon of the device
of FIG. 8A; and
[0047] FIG. 8C is a schematic effective circuit of the device of
FIG. 8A.
DETAILED DESCRIPTION
[0048] Developing a quantum computer involves a number of different
technical developments, some of which build upon each other. As an
initial step, a quantum system must be developed that can be
controlled sufficiently well to hold one bit of quantum information
(a `qubit`) long enough for the qubit to be written, manipulated,
and read. Once this has been achieved, quantum algorithms can be
performed on these quantum systems if a number of additional
requirements, known as the DiVincenzo criteria, are also satisfied.
One of these criteria is the ability to implement a universal set
of gates. That is, to implement gates that in combination can
realize complex quantum algorithms. Unlike in classical computing,
however, in which any desired Boolean gate can be implemented from
NAND (or NOR) gates alone, in a quantum computer universality can
only be achieved with a combination of arbitrary single qubit gates
and a two-qubit gate (e.g., a CNOT gate).
[0049] Another of the DiVincenzo criteria is to produce qubits that
have sufficiently long decoherence times to be able to perform
computation. Some techniques to help meet this criteria employ
quantum error correction techniques to correct decoherence errors
in a quantum system once they occur. If the error correction
operations are sufficiently effective, the state of a quantum
system can be maintained for a long time, and possibly
indefinitely.
[0050] The inventors have recognized and appreciated techniques for
implementing a universal set of quantum logic gates in a system
that satisfies the DiVincenzo criteria. Quantum information may be
stored in linear quantum mechanical oscillators that are coupled to
one another by a multi-level (e.g., nonlinear) quantum system. The
states of the linear quantum mechanical oscillators act a logical
qubit for storing a single bit of quantum information. By
controlling the quantum mechanical oscillators and the multi-level
quantum system with driving signals, a universal set of quantum
logic gates can be implemented. For example, arbitrary single qubit
rotations can be performed, as well as entangling and disentangling
operations between two or more qubits.
[0051] These techniques comprise an operation for generating an
entangled state between two quantum mechanical oscillators. Such a
state may enable logical operations between two logical qubits,
where each logical qubit is represented by a state of one of the
oscillators, and may further enable quantum error correction
techniques to be applied to these qubits. Accordingly, these
techniques may support the two DiVincenzo criteria discussed above,
by simultaneously (i) allowing logical operations to be performed
upon two qubits, and by (ii) lengthening decoherence times by
enabling quantum error correction techniques.
[0052] In some embodiments, a suitable device architecture may
include a multi-level quantum system, such as a transmon or other
nonlinear quantum system, dispersively coupled to two qubits each
implemented as a quantum mechanical oscillator. The oscillators may
be, for example, resonator cavities or other suitable linear
quantum oscillators. The multi-level quantum system may be used as
an ancilla to create, manipulate, and/or to measure the quantum
states of each of the oscillators to which it is coupled. By
accessing multiple energy levels of the ancilla, the techniques
described herein make it possible to realize universal quantum
control of the two qubits and to monitor the error syndrome of the
two qubits by performing quantum non-demolition (QND)
measurements.
[0053] A nonlinear quantum system is a quantum system that does not
have an infinite number of energy levels (e.g., energy eigenstates)
separated by a constant energy difference. In contrast, a linear
quantum system has an infinite number of evenly distributed energy
levels. An example of a linear quantum system is a quantum
mechanical oscillator. An example of a nonlinear quantum system is
a two-level quantum system (e.g., a two-level atom) which only has
two energy eigenstates. Another example of a nonlinear quantum
system is a multi-level quantum system such as a superconducting
qubit (e.g., a transmon).
[0054] Conventionally, nonlinear quantum systems are used to store
quantum information. For example, it has been shown that transmons
can be used to implement a qubit. The inventors, however, have
recognized and appreciated that storing quantum information in
linear quantum mechanical oscillators has several advantages over
storing the information in nonlinear quantum systems. One such
advantage is an increase in coherence time. In particular, the
inventors have recognized and appreciated that so-called "cat
states" may be a particularly useful type of state of the quantum
mechanical oscillators to which to apply the techniques described
herein.
[0055] A cat state is a coherent superposition of two coherent
states with opposite phase. For example, in a quantum harmonic
oscillator a cat state can be described by
1 2 ( .alpha. + - .alpha. ) , ##EQU00001##
where |.alpha. is a coherent state with a first phase and |-.alpha.
is a coherent state with a second phase that is shifted 180 degrees
relative to the first phase. At large |.alpha.|, the two components
of the cat state correspond to distinct quasi-classical
wave-packets, drawing analogy to Schrodinger's symbolic paradox of
an unfortunate cat inside a closed box being simultaneously dead
and alive. Cat states have so far been realized with single-mode
optical or microwave fields with up to about 100 photons, but are
increasingly susceptible to decoherence as the number state
increases in magnitude.
[0056] According to some embodiments, an entangled state across two
quantum mechanical oscillators may be produced by entangling cat
states of the oscillators. Techniques for producing such a state
are discussed below, but initially properties of the entangled
state will be described. The entangled state may be expressed
as:
.psi. .+-. = 1 2 ( .alpha. A .alpha. B .+-. - .alpha. A - .alpha. B
) ( Eqn . 1 ) ##EQU00002##
[0057] where |.+-..alpha..sub.A and |.+-..alpha..sub.B are coherent
states of two oscillator eigenmodes whose amplitudes are prepared
to be equal for convenience. The two oscillators are referred to
herein as "Alice" and "Bob." Each of the two modes are
predominantly localized in one of the two oscillators that are
weakly connected via the ancilla.
[0058] Despite a nonzero (but small) spatial overlap of the two
modes, for convenience we refer to the two modes herein as the
states of the two oscillators. For larger |.alpha.| (e.g.,
|.alpha.|.sup.2.gtoreq.2), |.psi..sub..+-. can be considered a
single cat state living in two boxes whose superposed components
are coherent states in a hybridized mode involving both Alice and
Bob. Alternatively, in the more natural eigenmode basis,
|.psi..sub..+-. may also be understood as two single-oscillator cat
states that are entangled with each other.
[0059] Multi-oscillator cat states can be a useful way of encoding
quantum information that allows fault-tolerant quantum computation,
where quantum information is redundantly encoded in the coherent
state basis of the multiple oscillators. In this context, the
techniques described herein realize an architecture of two coupled
logical qubits. The two-mode cat state can be considered a
two-qubit Bell state
1 2 ( 0 0 .+-. 1 1 ) ##EQU00003##
of the logical qubits, where the first quasi-orthogonal coherent
states |.alpha. represents the logical state |0 for each of the two
oscillators and the second quasi-othogonal coherent state |-.alpha.
represents the logical state |1 for each of the two
oscillators.
[0060] According to some embodiments, the quantum mechanical
oscillators may be bosonic systems. In such cases, the two-mode cat
state is an eigenstate of the joint boson number parity operator
P.sub.J:
P.sub.J=P.sub.AP.sub.E=e.sup.i.pi.a.sup..dagger..sup.ae.sup.i.pi.a.sup..-
dagger..sup.b (Eqn. 2)
where a(a.sup..dagger.) and b(b.sup..dagger.) are the annihilation
(creation) operators of bosons in Alice and Bob, and P.sub.A and
P.sub.B are the boson number parity operators on the individual
oscillators. Remarkably, |.psi..sub.+ (or |.psi..sub.-) has
definitively even (or odd) number of bosons in two cavities
combined, while the boson number parity in each cavity is maximally
uncertain. The error syndrome of the two-mode cat state may
therefore be monitored by performing quantum non-demolition (QND)
measurements of the cat state.
[0061] According to some embodiments, QND measurements of the joint
parity may be performed by probing the two-mode cat state via the
coupled ancilla. The results of such measurements, discussed below,
are not only illustrative of the highly non-classical properties of
the state, but also the fundamental tools for quantum error
correction in general. According to some embodiments, the system
may include a readout unit coupled to the ancilla (which is in turn
coupled to each of the two quantum mechanical oscillators). The
readout unit may be, for example, a resonating cavity that may be
used to projectively measure the ancilla state. The readout unit
may thereby provide for the above-mentioned QND measurements of the
two oscillators, including but not limited to joint and/or single
parity measurements of the oscillators.
[0062] Following below are more detailed descriptions of various
concepts related to, and embodiments of, techniques for techniques
for generating, manipulating and/or probing an entangled state
across two quantum mechanical oscillators. It should be appreciated
that various aspects described herein may be implemented in any of
numerous ways. Examples of specific implementations are provided
herein for illustrative purposes only. In addition, the various
aspects described in the embodiments below may be used alone or in
any combination, and are not limited to the combinations explicitly
described herein.
[0063] FIG. 1 is a block diagram of a circuit quantum
electrodynamics system suitable for practicing aspects of the
present disclosure. System 100 includes quantum mechanical
oscillators 110 ("Alice") and 120 ("Bob") dispersively coupled to a
multi-level quantum system 130 ("ancilla"). An electromagnetic
signal .epsilon..sub.A(t) may be applied to the oscillator 110, an
electromagnetic signal .epsilon..sub.B (t) may be applied to the
oscillator 120, and an electromagnetic signal
.epsilon..sub.ancilla(t) may be applied to the multi-level system
130. Generally in the discussion below, application of such an
electromagnetic signal or pulse may also be referred to as
"driving" of the oscillator or ancilla. In some embodiments, the
multi-level quantum system 130 may be a nonlinear quantum
system.
[0064] As discussed above, to manipulate the states of the two
oscillators, the multi-level quantum system 130 may be used as an
ancilla. One or more energy levels of the multi-level system may be
accessed in this process. For instance, the lowest two energy
levels, the lowest three energy levels, etc. or any other group of
energy levels may be accessed by .epsilon..sub.ancilla(t) to
produce interactions between the ancilla and the two oscillators
via the respective dispersive couplings, examples of which are
described below.
[0065] According to some embodiments, the Hamiltonian of system 100
which includes two oscillator modes, a multi-level system, and
their dispersive interactions can be written as:
H/
=.omega..sub.Aa.sup..dagger.a+.omega..sub.Bb.sup..dagger.b+.omega..su-
b.ge|ee|+(.omega..sub.ge+.omega..sub.ef)|ff|
-.chi..sub.A.sup.gea.sup..dagger.a|ee|-(.chi..sub.A.sup.ge+.chi..sub.A.s-
up.ef)a.sup..dagger.a|ff|
-.chi..sub.B.sup.geb.sup..dagger.b|ee|-(.chi..sub.B.sup.ge+.chi..sub.B.s-
up.ef)b.sup..dagger.b|ff| (Eqn. 3)
where a (a.sup..dagger.) and b(b.sup..dagger.) are the annihilation
(creation) operators of energy quanta in the oscillators Alice and
Bob, |g, |e and |f are the lowest three energy levels of the
ancilla, .omega..sub.A and .omega..sub.B are the angular
frequencies of the two oscillators (Alice and Bob), .omega..sub.ge
and .omega..sub.ef are the |e.fwdarw.|g and |f.fwdarw.|e transition
frequencies of the ancilla, and .chi..sub.i.sup.ge and
.chi..sub.i.sup.ef (i=A or B) represent the dispersive frequency
shifts of oscillator i associated with the two ancilla transitions.
Small higher-order nonlinearities are neglected in Eqn. 1 for
simplicity.
[0066] The time-dependent driving signals .epsilon..sub.A(t),
.epsilon..sub.B(t) and .epsilon..sub.ancilla(t), also referred to
herein as "drive waveforms," may be applied to Alice, Bob and the
ancilla, respectively, to realize arbitrary single qubit operations
upon each of these elements. One illustrative approach to
determining and applying these time-dependent drives is based on
Optimal Control Theory (OCT) and is described in International
Patent Application No. PCT/US16/43514, filed on Jul. 22, 2016,
titled "Techniques of Oscillator State Manipulation for Quantum
Information Processing and Related Systems and Methods," which is
hereby incorporated by reference in its entirety. Other examples of
apparatus and methods for applying control pulses to apply logic
gates to and perform other manipulations on a coupled
transmon/oscillator quibit system are described in U.S. Provisional
Patent Application No. 62/294,966, filed on Feb. 12, 2016, titled
"Quantum Computer State Controller," which is hereby incorporated
by reference in its entirety. Such quantum logic gates between
oscillator and ancilla states are the key tools for deterministic
generation and manipulation of the two-mode cat state
|.psi..sub..+-., for example, and for enabling
continuous-variable-based quantum computation.
[0067] The inventors have recognized and appreciated a process for
producing the two mode entangled cat state described above with
respect to the two oscillators of system 100 that may be broadly
described as follows. Initially, the multi-level quantum system 130
may be manipulated into a superposition of two energy levels. One
approach to produce this result may be to drive the ancilla with
.epsilon..sub.ancilla(t) to produce a rotation of the ancilla state
in the |g-|e Bloch sphere. Irrespective of how the ancilla is
arranged to be in this state, each oscillator may subsequently be
driven by a displacement conditional on the state of the ancilla,
which entangles each oscillator with the state of the ancilla. For
instance, if the conditional displacement is applied with each
oscillator in the |0 state and the displacement is conditional on
the ancilla being in |g, the displacement realizes a three-way
entangling gate:
1 2 ( g + e ) 0 A 0 B .fwdarw. 1 2 ( g 0 A 0 B + e 2 .alpha. A 2
.alpha. B ) ( Eqn . 4 ) ##EQU00004##
[0068] Subsequently, another rotation operation can be applied to
the ancilla that is conditional on the joint oscillator state,
which disentangles the ancilla and leaves the oscillators in a
two-mode cat state.
[0069] According to some embodiments, the state-dependent frequency
shifts (.chi.'s) of each oscillator with respect to each ancilla
transition are arranged to allow cavity state manipulations
conditioned on the ancilla level, or vice versa, using
spectrally-selective control pulses. In practice, such an
arrangement comprises forming the oscillators and ancilla system to
have different resonant frequencies that enabled such
manipulations. An example of one such arrangement is discussed
below.
[0070] This above-described cat state entanglement process is
illustrated in FIG. 2, which shows a control sequence for producing
cat states spanning modes of two quantum mechanical oscillators,
according to some embodiments. Control sequence 200 illustrates the
states of Alice, Bob and the Ancilla system with time running left
to right in the figure.
[0071] In illustrative control sequence 200, Alice, Bob and the
Ancilla system begin in distinct initial states. In some
embodiments, the three systems have initial states that are ground
states of the respective systems. For instance, Alice and Bob may
be in the |0 state and the ancilla may be in the |g state. While
these initial states may represent convenient starting points for
the illustrated control sequence, other initial states may be
contemplated so long as the described entanglement between cat
states can be produced.
[0072] In act 210, the ancilla is controlled to be in a
superposition of states. This may be achieved by driving the
ancilla with a driving signal to cause a rotation of the quantum
state of the ancilla on a Bloch sphere associated with two
eigenstates of the ancilla. The superposition may be a
superposition of any number of the energy levels of the multi-level
ancilla system, and any superposition of these energy levels may be
produced. The key steps in control sequence 200 are the conditional
displacements 220 and 221, which are conditional on the state of
the ancilla and produce entanglement between the ancilla and each
of the two oscillators. So long as these displacements may be made
conditional on states of the superposition of the ancilla produced
in act 210 to produce such entangelment, any suitable superposition
may be produced in act 210.
[0073] In acts 220 and 221, gates are applied to Alice and Bob,
respectively, that coherently add (or remove) energy to (or from)
the oscillator conditional on the state of the coupled ancilla.
Since the ancilla is in a superposition of states at this stage,
making the displacements 220 and 221 conditional on at least one of
those states of the superposition produces a superposition of
states in the respective oscillator that is entangled with the
state of the ancilla.
[0074] In optional act 230, a rotation may be applied to the
ancilla conditional on the states of Alice and Bob. This rotation
may disentangle the ancilla from the oscillators, yet may leave the
oscillators entangled with one another via their (weak) couplings
through the ancilla.
[0075] To illustrate one possible experimental realization of
system 100 shown in FIG. 1, FIGS. 3A and 3B depict an illustrative
circuit quantum electrodynamics (cQED) system comprising two
coaxial resonator cavities coupled to an ancilla transmon,
according to some embodiments. In system 300, which is a
three-dimensional schematic of the device shown in FIG. 3A,
resonators 310 and 320 function as quantum mechanical oscillators
110 and 120 in the system of FIG. 1, and transmon 330 functions as
the multi-level quantum system 130 in the system of FIG. 1. The
resonators 310 and 320, as illustrated, are coaxial cavities formed
from aluminum.
[0076] In the example of FIGS. 3A-3B, the cQED system also includes
a quasi-planar linear resonator that may be operated to readout the
state of the ancilla transmon. The amplitude and phase of the pulse
produced by the readout resonator near its resonance frequency both
depend on the quantum state of the ancilla transmon. The resonator
is formed by the resonator cavity shown in FIG. 3A and by the
stripline on the transmon chip which is the central element of the
coaxial arrangement shown. The transmon features three antennas
331, 332 and 333, which are coupled to the cavity 310, cavity 320
and readout resonator 340, respectively. Josephson junction 335 is
coupled to each of the antennas.
[0077] The illustrative cQED system 300 shown in FIGS. 3A-3B is
explored in more detail below, with FIG. 4 depicting a specific
control sequence for producing a two-mode cat state across the
resonator cavities 310 and 320; FIG. 5 depicting an illustrative
technique for effectively producing a conditional displacement on
the resonator cavities by applying two non-conditional
displacements with a waiting period in between; FIGS. 6A-6B
depicting two illustrative control sequences for experimentally
measuring the joint parity of the two resonator cavities; FIG. 7
illustrates an experimental setup for operation of system 300; and
FIGS. 8A-8C depict a physical implementation of system 300 from a
block of high-purity aluminum.
[0078] While many experimental implementations and configurations
may be envisioned based on the type of cQED system shown in FIGS.
3A-3B, one illustrative configuration in terms of the properties of
each resonator and the transmon will be described. Table 1 below
shows the Hamiltonian parameters of each component of an
illustrative embodiment of system 300, including the transmon
ancilla, the two cavity resonators (Alice and Bob) and the readout
resonator. The measured parameters include all transition
frequencies (.omega./2.pi.), dispersive shifts between each
resonator and each transmon transition (.chi./2.pi.), the self-Kerr
of Alice (K.sub.A/2.pi.) and Bob (K.sub.B/2.pi.), and the
cross-Kerr interaction between Alice and Bob (K.sub.AB/2.pi.). The
Kerr parameters and .chi..sub.d.sup.ef associated with the readout
resonator are theoretical estimates based on the other measured
parameters.
TABLE-US-00001 TABLE 1 Nonlinear interactions Frequency Alice Bob
versus: .omega./2.pi. X.sub.A/2.pi. X.sub.B/2.pi. Readout |e
.fwdarw.|g 4.87805 GHz 0.71 MHz 1.41 MHz 1.74 MHz |f .fwdarw.|e
4.76288 GHz 1.54 MHz 0.93 MHz 1.63 MHz Alice 4.2196612 GHz 0.83 kHz
-9 kHz 5 kHz Bob 5.4467677 GHz -9 kHz 5.6 kHz 12 kHz Readout 7.6970
GHz 5 kHz 12 kHz 7 kHz
[0079] In the example of Table 1, it will be noted that
.omega..sub.A<.omega..sub.ef<.omega..sub.ge<.omega..sub.B.
As discussed below in relation to FIGS. 6A-6B, this arrangement
allows for measurements of the joint parity of the resonators
without the requirement that .chi..sub.A.sup.ge is exactly equal to
.chi..sub.B.sup.ge.
[0080] FIG. 4 illustrates a control sequence suitable for producing
cat states spanning modes of two quantum mechanical resonators,
according to some embodiments. Control sequence 400 may be applied
to system 300 shown in FIGS. 3A-3B, for example.
[0081] In the example of FIG. 4, resonators Alice and Bob and the
Ancilla transmon, are initially in ground states |0 state and |g,
respectively. In act 410, an ancilla superposition is prepared by
performing a rotation R.sub..pi./2.sup.ge that is an ancilla
rotation of .pi./2 in the X-Y plane of the Bloch sphere of the
|g-|e manifold. This places the ancilla in an equal superposition
of the ground and excited states,
1 2 ( g + e ) . ##EQU00005##
[0082] In act 420, conditional displacements D.sub.2.alpha..sup.g
are applied to each of the two cavities. By applying the
time-dependent microwave control pulses .epsilon..sub.A(t) and
.epsilon..sub.B(t) to the cavities, arbitrary cavity state
displacements may be produced in Alice
(D.sub..beta..sub.A=e.sup..beta..sup.A.sup.a.sup..dagger..sup.-.beta.*.su-
p.A.sup.a) and Bob
(D.sub..beta..sub.B=e.sup..beta..sup.B.sup.b.sup..dagger..sup.-.beta.*.su-
p.B.sup.b) independently. In the example of FIG. 4, the conditional
displacements are operations that would put the respective
resonators in the |2.alpha. state (add energy to move the state
from |0 to |2.alpha.) when the coupled ancilla transmon is in the
|g state. The net result of these displacements is to realize a
three-way entangling gate:
1 2 ( g + e ) 0 A 0 B .fwdarw. 1 2 ( g 0 A 0 B + e 2 .alpha. A 2
.alpha. B ) ##EQU00006##
[0083] In act 430, an ancilla rotation (R.sub..pi..sup.00)
conditional on the cavity state |0.sub.A|10.sub.B disentangles the
ancilla and leaves the cavities in a two-mode cat state. That is,
the ancilla returns to |g yet the cavities remain in the
1 2 ( 0 A 0 B + 2 .alpha. A 2 .alpha. B ) ##EQU00007##
state.
[0084] In act 440, additional displacements D.sub.-.alpha. are
applied to the cavities. These are unconditional displacements of
Alice and Bob (D.sub.-.alpha.) to center the cat state in the phase
space. This is a trivial step purely for the convenience of the
presentation, and produces a cat state:
1 2 ( - .alpha. A - .alpha. B + .alpha. A .alpha. B ) .
##EQU00008##
[0085] As discussed above, the conditional displacement
(D.sub.2.alpha..sup.g) is what allows the entangling of the ancilla
with the cavities, and therefore allows the generation of an
entangled state between the two cavities. This operation can be
directly implemented using cavity drives with a bandwidth smaller
than the dispersive interaction strength (.chi..sub.i.sup.ge, i=A
or B). However, this method requires a comparatively long pulse
duration (and therefore higher infidelity due to decoherence and
Kerr effects). An alternative method for producing the conditional
displacement (D.sub.2.alpha..sup.g) is shown in FIG. 5.
[0086] FIG. 5 illustrates a control sequence in which the
conditional displacement D.sub.2.alpha..sup.g is effectively
realized by two unconditional displacements separated by a wait
time .DELTA.t in between. In the example of FIG. 5, steps 510, 530
and 540 are equivalent to steps 410, 430 and 440 shown in FIG. 4.
The difference between FIG. 4 and FIG. 5 is that steps 521, 522 and
523 together effectively implement step 420 in FIG. 4--that is, the
conditional displacement D.sub.2.alpha..sup.g.
[0087] In step 521, non-conditional displacements
D.sub..alpha..sub.1 and D.sub..alpha..sub.2 are applied to Alice
and Bob, respectively, resulting in the product state of the two
cavities being |.alpha..sub.1 |.alpha..sub.2. In some embodiments,
the two displacements may have equal magnitude and phase (e.g.,
.alpha..sub.1=.alpha..sub.2). In step 522, a wait time .DELTA.t is
performed, during which time the states in Alice and Bob accumulate
conditional phases that are dependent on the state of the ancilla,
represented by the unitary operator U(t). Specifically, during the
wait time .DELTA.t, due to the dispersive interactions between each
cavity and the ancilla, cavity coherent states in both cavities
accumulate conditional phases of
.PHI..sub.i=.chi..sub.i.sup.ge.DELTA.t if the ancilla is in |e:
U(.DELTA.t)=|.sub.A|.sub.B|gg|+e.sup.i.PHI..sup.A.sup.a.sup..dagger..sup-
.ae.sup.i.PHI..sup.B.sup.b.sup..dagger..sup.b|ee| (Eqn. 5)
[0088] The net result is that the states of the cavities evolve
into a three-way entangled state in the manner:
1 2 ( g + e ) .alpha. 1 .alpha. 2 .fwdarw. 1 2 ( g .alpha. 1
.alpha. 2 + e .alpha. 1 e i .phi. A .alpha. 2 e i .phi. B )
##EQU00009##
[0089] An additional displacement applied in act 523 to each cavity
after this time evolution step effectively arrives at the same
result as the conditional displacement described above. Using the
IQ plane to describe the photon probability distribution in each
cavity in the rotating frame, a coherent state |.alpha.'.sub.i can
be represented by a (Gaussian) circle that stays stationary when
the ancilla is in |g, and rotates with the angular velocity
.chi..sub.i.sup.ge when the ancilla is in |e:
|.alpha.'.sub.i.fwdarw.|.alpha.'e.sup.i.chi..sup.i.sup.ge.sup..DELTA.t.su-
b.i. Therefore, this conditional phase gate can split cavity
coherent state in the phase space when the ancilla is prepared
in
1 2 ( g + e ) , ##EQU00010##
effectively realizing a conditional displacement.
[0090] FIGS. 6A-6B are illustrative control sequences depicting two
approaches to experimentally measuring the joint parity of two
quantum mechanical oscillators, according to some embodiments. As
discussed above, measurement of joint parity number is important
both to understand the two-mode cat state and to be able to detect
error syndromes and correct them to maintain the state via quantum
error correction techniques.
[0091] For the illustrative system 300 shown in FIGS. 3A-3B, the
joint parity measurement is a joint photon number parity, P.sub.J,
measurement. This single-cavity photon parity measurements using
only the only |g and |e levels of an ancilla qubit is applicable to
one cavity when the other cavity is in the vacuum state. The
measurement uses the dispersive interaction .chi..sub.i.sup.ge to
map even-photon-number and odd-photon-number states in the cavity
of interest (i=A or B) to different levels of the ancilla. This may
be realized by two .pi./2 rotations of the ancilla qubit,
R.sub..pi./2.sup.ge (around the same axis, e.g., the X-axis),
separated by a wait time of .pi./.chi..sub.i.sup.ge. For example,
if Bob is in the vacuum state (b.sup..dagger.b|0=0), the
conditional phase shift described in Eqn. 5 over the time
.DELTA.t=.pi./.chi..sub.A.sup.ge is described by the following
unitary operator:
U(.pi./.chi..sub.A.sup.ge)=C.sub..pi..sup.A=||gg|+e.sup.i.pi.a.sup..dagg-
er..sup.a|ee| (Eqn. 6)
[0092] This shift is equivalent to a qubit Z-rotation of .pi.
conditioned on the photon number in Alice being odd because
e.sup.i.pi.a.sup..dagger..sup.a=P.sub.A. Therefore the whole
sequence R.sub..pi./2.sup.geC.sub..pi..sup.AR.sub..pi./2.sup.ge
flips the qubit if and only if the photon number parity in Alice is
even, and therefore a subsequent readout of the qubit state
measures the parity of the cavity.
[0093] This control and measurement sequence to measure parity in a
single cavity can in principle be implemented to also measure the
joint photon number parity, but only if .chi..sub.A.sup.ge is
exactly equal to .chi..sub.B.sup.ge. This is because for a wait
time of .DELTA.t=.pi./.chi..sub.A.sup.ge(=.pi./.chi..sub.B.sup.ge),
from Eqn. 5 we have:
U(.pi./.chi..sub.i.sup.ge)=C.sub..pi..sup.AC.sub..pi..sup.B=||gg|+P.sub.-
AP.sub.B|ee| (Eqn. 7)
[0094] Noting P.sub.J, =P.sub.AP.sub.B, an identical control
sequence of R.sub..pi./2.sup.geU(.DELTA.t)R.sub..pi./2.sup.ge
followed by a qubit readout would achieve the joint parity
measurement. However, without strictly identical .chi..sub.A.sup.ge
and .chi..sub.B.sup.ge, the phase accumulation in one cavity is
faster than the other, and it is in general not possible to realize
parity operators in both cavities simultaneously using this simple
protocol. Moreover, for a general two-cavity quantum state, this
sequence cannot measure single-cavity parity operator (P.sub.A or
P.sub.B) due to inevitable entanglement between the ancilla and the
photons in the other cavity during the process.
[0095] According to some embodiments, one technique for measuring
P.sub.J, with a less stringent requirement on Hamiltonian
parameters works by exploiting the |f-level of the ancilla. This
method is may be advantageous when |e.fwdarw.|g transition of the
ancilla shows stronger interaction with Bob
(.chi..sub.B.sup.ge>.chi..sub.A.sup.ge), while the |f.fwdarw.|e
transition shows stronger interaction with Alice
(.chi..sub.A.sup.ef>.chi..sub.B.sup.ef). This can be physically
realized by engineering the ancilla frequency between the two
cavities, i. e.
.omega..sub.A<.omega..sub.ef<.omega..sub.ge<.omega..sub.B.
[0096] Considering the quantum state with two cavities and three
ancilla levels in general, the unitary evolution for any wait time
.DELTA.t is:
U ( .DELTA. t ) = A B g g + e i .phi. A a .dagger. a e i .phi. B b
.dagger. b e e + e i .phi. A ' a .dagger. a e i .phi. B ' b
.dagger. b f f ( Eqn . 8 ) ##EQU00011## where
.PHI..sub.A=.chi..sub.A.sup.ge.DELTA.t,.PHI..sub.B=.chi..sub.B.sup.ge.DE-
LTA.t
.PHI.'.sub.A=.chi..sub.A.sup.gf.DELTA.t,.PHI.'.sub.B=.chi..sub.B.sup.gf.-
DELTA.t (Eqn. 9)
[0097] Here we define
.chi..sub.A.sup.gf.ident..chi..sub.A.sup.ge+.chi..sub.A.sup.ef and
.chi..sub.B.sup.gf.ident..chi..sub.B.sup.ge+.chi..sub.B.sup.ef.
Therefore, the two cavities simultaneously acquire conditional
phases in their coherent state components at relative rates that
differ for |e and |f. FIGS. 6A and 6B illustrate two different
pulse sequences for realizing a joint parity measurement without
the requirement that .chi..sub.A.sup.ge and .chi..sub.B.sup.ge are
equal. In both FIG. 6A and FIG. 6B, step 610 represents the
generation of a state between two oscillators (Alice and Bob) and
an ancilla. Step 610 may comprise, for instance, control sequence
200 shown in FIG. 2 or control sequence 400 shown in FIG. 4. Step
620 in each of FIG. 6A and FIG. 6B are initial displacements of the
cavities.
[0098] In the example of FIG. 6A, for a given two-cavity quantum
state .PSI..sub.AB, we can first use a R.sub..pi./2.sup.ge rotation
in act 630 to prepare the ancilla in the state
1 2 ( g + e ) . ##EQU00012##
Then, a wait time .DELTA.t.sub.1 in act 640 imparts phases
.PHI..sub.A1=.chi..sub.A.sup.ge.DELTA.t.sub.1 and
.PHI..sub.B1=.chi..sub.B.sup.ge.DELTA.t.sub.1 to the two cavities
for the |e component of the state:
.PSI. AB 1 2 ( g + e ) 1 2 [ .PSI. AB g + e i .phi. A 1 a .dagger.
a e i .phi. B 1 b .dagger. b .PSI. AB e ] ( Eqn . 10 )
##EQU00013##
[0099] Next, in act 650 the |e component in this intermediate state
is converted to |f by a .pi. rotation in the |e-|f space,
R.sub..pi..sup.ef. Subsequently, a second wait time .DELTA.t.sub.2
in act 660 leads to a second simultaneous conditional phase gate,
imparting phases .PHI..sub.A2=.chi..sub.A.sup.gf.DELTA.t.sub.2 and
.PHI..sub.B2=.chi..sub.B.sup.gf.DELTA.t.sub.2 to the two cavities
for the now |f component of the state:
1 2 [ .PSI. AB g + e i .phi. A 1 a .dagger. a e i .phi. B 1 b
.dagger. b .PSI. AB f ] 1 2 [ .PSI. AB g + e i ( .phi. A 1 + .phi.
A 2 ) a .dagger. a e i ( .phi. B 1 + .phi. B 2 ) b .dagger. b .PSI.
AB f ] ( Eqn . 11 ] ##EQU00014##
[0100] The |f component is then converted back to |e by another
R.sub..pi..sup.ef pulse in act 670. When .DELTA.t.sub.1 and
.DELTA.t.sub.2 satisfy the following equations:
.PHI..sub.A1+.PHI..sub.A2=.chi..sub.A.sup.ge.DELTA.t.sub.1+.chi..sub.A.s-
up.gf.DELTA.t.sub.2=.pi.
.PHI..sub.B1+.PHI..sub.B2=.chi..sub.B.sup.ge.DELTA.t.sub.1+.chi..sub.B.s-
up.gf.DELTA.t.sub.2=.pi. (Eqn. 12)
the obtained quantum state is:
1 2 [ .PSI. AB g + P J .PSI. AB e ] ( Eqn . 13 ] ##EQU00015##
effectively realizing the simultaneous controlled .pi.-phase gate
(C.sub..pi..sup.AC.sub..pi..sup.B) in Eqn. 7. Finally, a
R.sub..pi./2.sup.ge pulse in act 680 completes the projection of
joint parity to the ancilla |g, |e levels, ready for readout
through the readout resonator in act 690.
[0101] The condition for finding non-negative solutions for
.DELTA.t.sub.1 and .DELTA.t.sub.2 in Eqn. 12 is that
.chi..sub.A.sup.ge-.chi..sub.B.sup.ge and
.chi..sub.A.sup.gf-.chi..sub.B.sup.gf have opposite signs. In
essence, the cavity that acquires phase slower than the other at |e
due to smaller .chi..sub.i.sup.ge is allowed to catch up at |f
using its larger .chi..sub.i.sup.gf.
[0102] It should be noted that such relative relation of .chi.'s is
just a practically preferred condition rather than an absolute
mathematical requirement. This is because parity mapping can be
achieved whenever both cavities acquire a conditional phase of .pi.
modulo 2.pi.. It is always possible to allow extra multiples of
2.pi. phases applied to the cavity with stronger dispersive
coupling to the ancilla, although it increases the total gate time
and incurs more decoherence. The most important ingredient in
engineering the P.sub.J operator is the extra tuning parameter
.DELTA.t.sub.2 (in addition to .DELTA.t.sub.1) that allows two
equations such as Eqn. 12 to be simultaneously satisfied.
[0103] This extra degree of freedom also enables measurement of
photon number parity of a single cavity, P.sub.A or P.sub.B, for an
arbitrary two-cavity quantum state. This alternative can be
realized with the same control sequences as shown in FIG. 6A whilst
choosing such wait times that one cavity acquires a conditional
.pi. phase (modulo 2 .pi.) while the other acquires 0 phase (modulo
2 .pi.). For example, to measure P.sub.A we use .DELTA.t.sub.1 and
.DELTA.t.sub.2 satisfying:
.PHI..sub.A1+.PHI..sub.A2=.chi..sub.A.sup.ge.DELTA.t.sub.1+.chi..sub.A.s-
up.gf.DELTA.t.sub.2=.pi.(mod 2.pi.)
.PHI..sub.B1+.PHI..sub.B2=.chi..sub.B.sup.ge.DELTA.t.sub.1+.chi..sub.B.s-
up.gf.DELTA.t.sub.2=.pi.(mod 2.pi.) (Eqn. 14)
[0104] FIG. 6B shows this alternative version of joint parity
mapping protocol, which uses more ancilla operations, but is more
adaptive to a larger parameter space of .chi.'s. In this protocol,
the ancilla spends time at |e-|f superposition, when conditional
phases proportional to .chi..sub.i.sup.ef are applied to the
cavities. To achieve joint parity mapping, the two time intervals
.DELTA.t.sub.1 and .DELTA.t.sub.2 therefore should satisfy:
.PHI..sub.A1+.PHI..sub.A2=.chi..sub.A.sup.ef.DELTA.t.sub.1+.chi..sub.A.s-
up.gf.DELTA.t.sub.2=.pi.(mod 2.pi.)
.PHI..sub.B1+.PHI..sub.B2=.chi..sub.B.sup.ef.DELTA.t.sub.1+.chi..sub.B.s-
up.gf.DELTA.t.sub.2=.pi.(mod 2.pi.) (Eqn. 15)
which can avoid the use of extra 2.pi. phases when
.chi..sub.A.sup.ef-.chi..sub.B.sup.ef has opposite sign versus
.chi..sub.A.sup.gf-.chi..sub.B.sup.gf.
[0105] Experimentally, choices of which parity mapping sequence to
apply (FIG. 6A or FIG. 6B) and gate times .DELTA.t.sub.1 and
.DELTA.t.sub.2 involve trade-offs in various aspects such as pulse
speed/bandwidth and coherence time. For the sequence of FIG. 6A,
.DELTA.t.sub.1=0, .DELTA.t.sub.2=184 ns has been experimentally
implemented. For the sequence of FIG. 6B, .DELTA.t.sub.1=28 ns,
.DELTA.t.sub.2=168 ns has been experimentally implemented. The
actual effective wait time is longer due to the non-zero duration
(16 ns) of each ancilla rotation. The first protocol, with this
choice of wait times, does not yield the exact .pi. phases required
for exact parity mapping (We estimate
.PHI..sub.A1+.PHI..sub.A2=0.97.pi. and
.PHI..sub.B1+.PHI..sub.B2=1.03.pi.. These phase errors lead to an
estimated infidelity of the joint parity measurement of about 3%
for the two-cavity states under this study. Exact phases can be
achieved with longer wait times so that
.PHI..sub.A1+.PHI..sub.A2=3.pi. and
.PHI..sub.B1+.PHI..sub.B2=5.pi., but the infidelity due to
decoherence and high-order Hamiltonian terms outweighs the
benefits. In principle, the second protocol that achieves exact
.pi. phases at relatively short total gate time should be more
advantageous. However, using the second protocol, we observe
visibly identical results of joint Wigner tomography of the
two-mode cat states with fidelity nearly equal to the first
protocol. This may be attributed to extra infidelity from the more
complicated ancilla rotations involved in the second protocol.
[0106] FIG. 7 is a circuit diagram of an illustrative experimental
setup for controlling and/or measuring a system of two quantum
mechanical oscillators coupled to an ancilla multi-level quantum
system, according to some embodiments. System 700 includes a cQED
system 720 which may, for example, be cQED system 300 shown in
FIGS. 3A-3B or some other cQED system suitable for practicing the
techniques described herein.
[0107] System 700 includes three temperatures stages at 15 mK, 4K
and 300K, where the cQED system 720 is operated at the 15 mK stage.
For instance, the cQED system 720 may be installed inside a
Cryoperm magnetic shield and thermalized to the mixing chamber of a
dilution refrigerator with a base temperature of 15 mK. Low-pass
filters and infrared (eccosorb) filters may be used to reduce stray
radiation and photon shot noise. A Josephson parametric converter
(JPC) 730 is also mounted to the 15 mK stage, connected to the
output port of the device package via circulators, providing
near-quantum-limited amplification.
[0108] In the example of FIG. 7, a field programmable gate array
(FPGA) 740 operates both the quantum-control pulse sequences and
the data acquisition process. In some cases, the FPGA may access
stored waveforms for application to Alice, Bob and the Ancilla
(e.g., for performing rotations, displacements, etc. as described
above). In other cases, the FPGA may be programmed to compute
rather than store waveforms in real time. This latter approach may
lead to lower (or minimal) usage of waveform memory for a large
number of different cavity displacements, allowing many
measurements in a single run.
[0109] In the example of FIG. 7, cavity drives and transmon drives
are generated by sideband-modulation of continuous-wave (CW)
carrier tones produced by respective microwave generators. The
drive waveforms may be applied to each cavity and to the transmon
independently. The 4 FPGA analog channels are used as 2 IQ-pairs in
unit 750 that each control a cavity drive to implement arbitrary
cavity displacements. Rotations of the transmon ancilla are
controlled by another pair of IQ channels provided by an arbitrary
waveform generator (AWG 760) synchronized to the FPGA via a digital
marker. This IQ pair controls both |g_|e and |e_|f transitions by
using different intermediate frequencies (IF).
[0110] In the example of FIG. 7, ancilla readout may be performed
by heterodyne measurement of the microwave transmission of a
readout pulse through the two ports of the quasi-planar readout
resonator near its resonance frequency. Using a cQED dispersive
readout, the amplitude and phase of the transmitted signal depends
on the quantum state of the ancilla. This readout pulse is produced
by a microwave generator (RO 771) gated by a FPGA digital channel.
The transmitted signal, after amplified by the JPC, is further
amplified by a high electron mobility transistor (HEMT) at 4K and a
regular RF amplifier at room temperature. The amplified signal is
then mixed down to 50 MHz with the output of a "local oscillator"
(LO 772) microwave generator, and analyzed by the FPGA. A split
copy of the readout pulse is directly mixed with the LO without
entering the refrigerator to provide a phase reference for the
measured transmission.
[0111] According to some embodiments, the long lifetimes of the
cavities of the cQED system may allow preparation of highly
coherent cavity quantum states, but may also severely limit the
rate at which one can repeat the measurement process. (With
T.sub.1.apprxeq.3 ms for Alice, it takes 15-20 ms for the cavity
photon number to naturally decay to the order of 0.01.) Since
tomographic measurement of the two-cavity quantum state can require
large amounts of measurements, in some cases a four-wave mixing
process may be implemented to realize fast reset for both cavities.
These processes can effectively convert photons in Alice or Bob
into photons in the short-lived readout resonator mode using three
parametric pumping tones. For instance, this reset operation could
be applied for 400 .mu.s, and then experimental data can be
acquired with a repetition cycle of about 900 .mu.s.
[0112] FIGS. 8A-8C depict one illustrative physical implementation
of cQED system 300 formed from a block of high-purity aluminum.
FIG. 8A is a photograph of a machined aluminum package containing
two coaxial stub cavity resonators and a transmon. In the example
of FIG. 8A, a single block of high-purity (5N5) aluminum has been
machined to form a 3D structure that contains both superconducting
cavity resonators, and also functions as a package for a sapphire
chip with deposited Josephson junction.
[0113] Each of the two cavities in FIG. 8A can be considered a 3D
version of a) .lamda./4 transmission line resonator between a
center stub (e.g., 3.2 mm in diameter) and a cylindrical wall
(outer conductor) (e.g., 9.5 mm in diameter). The heights of the
stubs control the resonance frequency, and in the pictured example
are about 12.2 mm and 16.3 mm for Alice and Bob, respectively. A
tunnel (in the example having a maximum width of 5.8 mm and a
maximum height of 3.9 mm) is opened from the outside towards the
middle wall between the two cavities, creating a three way joint
between the tunnel and the two cavities. The whole package is
chemically etched by about 80 .mu.m after machining to improve the
surface quality of the cavity resonators.
[0114] In the example of FIGS. 8A-8C, the superconducting transmon
is on a 5.5 mm.times.27.5 mm chip, which is diced from a 430
.mu.m-thick c-plane sapphire wafer after fabrication. The transmon
on the chip is shown in FIG. 8B. The fabrication process used
electron-beam lithography and shadow-mask evaporation of an
Al/AlOx/Al Josephson junction. The sapphire chip is inserted into
the tunnel, with the antenna pads of the transmon slightly
intruding into the coaxial cavities to provide mode coupling. The
chip is mechanically held at one end with an aluminum clamping
structure and indium seal.
[0115] According to some embodiments, during the transmon
fabrication process, a 100 .mu.m.times.9.8 mm strip of aluminum
film may be deposited on the sapphire chip to form a readout
resonator. This metal strip and the wall of the tunnel form a
planar-3D hybrid .lamda./2 stripline resonator. This resonator
design has the advantages of both lithographic dimensional control
and low surface/radiation loss. Here, it is capacitively coupled to
the transmon, and strongly coupled to a 50.OMEGA. transmission line
for readout, as shown in FIG. 8C.
[0116] Having thus described several aspects of at least one
embodiment of this invention, it is to be appreciated that various
alterations, modifications, and improvements will readily occur to
those skilled in the art.
[0117] Such alterations, modifications, and improvements are
intended to be part of this disclosure, and are intended to be
within the spirit and scope of the invention. Further, though
advantages of the present invention are indicated, it should be
appreciated that not every embodiment of the technology described
herein will include every described advantage. Some embodiments may
not implement any features described as advantageous herein and in
some instances one or more of the described features may be
implemented to achieve further embodiments. Accordingly, the
foregoing description and drawings are by way of example only.
[0118] According to some aspects, a method is provided of
performing a quantum logic gate between a first logical qubit
implemented using a first quantum mechanical oscillator and a
second logical qubit implemented using a second quantum mechanical
oscillator, wherein a nonlinear quantum system is dispersively
coupled to the first quantum mechanical oscillator and the
nonlinear quantum system is dispersively coupled to the second
quantum mechanical oscillator, the method comprising applying a
first electromagnetic pulse to the nonlinear quantum system,
applying a second electromagnetic pulse to the first quantum
mechanical oscillator, and applying a third electromagnetic pulse
to the second quantum mechanical oscillator.
[0119] According to some embodiments, the quantum logic gate is an
entangling gate that entangles the first logical qubit and the
second logical qubit.
[0120] According to some embodiments, applying a first
electromagnetic pulse to the nonlinear quantum system comprises
causing a rotation of a state of the nonlinear quantum system, the
rotation operation configured to produce a superposition of two
energy eigenstates of the nonlinear quantum system, applying a
second electromagnetic pulse to the first quantum mechanical
oscillator comprises causing a first displacement on the first
quantum mechanical oscillator that displaces a state of the first
quantum mechanical oscillator by a first phase and a first
magnitude, and applying a third electromagnetic pulse to the second
quantum mechanical oscillator comprises causing a second
displacement on the second quantum mechanical oscillator that
displaces a state of the second quantum mechanical oscillator by a
second phase and a second magnitude.
[0121] According to some embodiments, the method further comprises
applying a fourth electromagnetic pulse to the nonlinear quantum
system to cause a conditional rotation of the state of the
nonlinear quantum system that is based on the state of the state of
the first quantum mechanical oscillator and the state of the second
quantum mechanical oscillator.
[0122] According to some embodiments, the first displacement
operation is a conditional displacement operation that displaces
the state of the first quantum mechanical oscillator based on the
state of the nonlinear quantum system, and the second displacement
operation is a conditional displacement operation that displaces
the state of the second quantum mechanical oscillator based on the
state of the nonlinear quantum system.
[0123] According to some embodiments, the first displacement
operation is a non-conditional displacement operation that
displaces the state of the first quantum mechanical oscillator
independent from the state of the nonlinear quantum system, and the
second displacement operation is a non-conditional displacement
operation that displaces the state of the second quantum mechanical
oscillator independent of the state of the nonlinear quantum
system, and the method further comprises waiting for a first time,
after performing the first displacement operation and the second
displacement operation, the waiting for the first time causing the
state of the first quantum mechanical oscillator to accumulate a
first phase based on the state of the nonlinear quantum system and
causing the state of the second quantum mechanical oscillator to
accumulate a second phase based on the state of the nonlinear
quantum system, wherein the first phase is different from the
second phase.
[0124] According to some embodiments, the method further comprises
applying a fifth electromagnetic pulse to the first quantum
mechanical oscillator to cause a third displacement on the first
quantum mechanical oscillator that displaces the state of the first
quantum mechanical oscillator by a third phase and a third
magnitude, wherein the third phase is different from the first
phase and/or the third magnitude is different from the first
magnitude, and applying a sixth electromagnetic pulse to the second
quantum mechanical oscillator to cause a fourth displacement on the
second quantum mechanical oscillator that displaces the state of
the second quantum mechanical oscillator by a fourth phase and a
fourth magnitude, wherein the fourth phase is different from the
second phase and/or the fourth magnitude is different from the
second magnitude.
[0125] According to some embodiments, performing the quantum logic
gate comprises performing a joint measurement gate that measures a
joint property of a state of the first quantum mechanical
oscillator and a state of the second quantum mechanical
oscillator.
[0126] According to some embodiments, performing a joint
measurement gate comprises mapping the joint property to a state of
the nonlinear system.
[0127] According to some embodiments, the joint property is a joint
parity, and wherein the mapping the joint property to the state of
the nonlinear system comprises applying a first electromagnetic
pulse to the nonlinear quantum system to cause a first rotation of
the state of the nonlinear quantum system, the first rotation being
a rotation on a Bloch sphere of a first manifold formed by a first
energy level of the nonlinear quantum system and a second energy
level of the nonlinear quantum system, applying a second
electromagnetic pulse to the first quantum mechanical oscillator to
impart a first conditional phase on the state of the first quantum
mechanical oscillator based on the state of the nonlinear quantum
system, applying a third electromagnetic pulse to the second
quantum mechanical oscillator to impart a second conditional phase
on the state of the second quantum mechanical oscillator based on
the state of the nonlinear quantum system, applying a fourth
electromagnetic pulse to the nonlinear quantum system to cause a
second rotation of the state of the nonlinear quantum system, the
second rotation being a rotation on a Bloch sphere of a second
manifold formed by the second energy level of the nonlinear quantum
and a third energy level of the nonlinear quantum, applying a fifth
electromagnetic pulse to the first quantum mechanical oscillator to
impart a third conditional phase on the state of the first quantum
mechanical oscillator based on the state of the nonlinear quantum
system, applying a sixth electromagnetic pulse to the second
quantum mechanical to impart a fourth conditional phase on the
state of the second quantum mechanical oscillator based on the
state of the nonlinear quantum system, applying a seventh
electromagnetic pulse to the nonlinear quantum system to cause a
third rotation on the state of the nonlinear quantum system, the
third rotation being a rotation on the Bloch sphere of the second
manifold formed by the second energy level and the third energy
level, and applying an eighth electromagnetic pulse to the
nonlinear quantum system to cause a fourth rotation on the state of
the nonlinear quantum system, the fourth rotation being a rotation
on the Bloch sphere of the first manifold formed by the first
energy level and the second energy level.
[0128] According to some embodiments, a first dispersive coupling
between the nonlinear quantum system and the first quantum
mechanical oscillator is fixed while performing the quantum logic
gate, a second dispersive coupling between the nonlinear quantum
system and the second quantum mechanical oscillator is fixed while
performing the quantum logic gate, and the first quantum mechanical
oscillator is not directly coupled to the second quantum mechanical
oscillator.
[0129] According to some embodiments, the first dispersive coupling
being fixed is a result of the nonlinear quantum system being
physically stationary relative to the first quantum mechanical
oscillator while performing the quantum logic gate and the resonant
frequency of the first quantum mechanical oscillator being fixed
while performing the quantum logic gate, and the second dispersive
coupling being fixed is a result of the nonlinear quantum system
being physically stationary relative to the second quantum
mechanical oscillator while performing the quantum logic gate and
the resonant frequency of the second quantum mechanical oscillator
being fixed while performing the quantum logic gate.
[0130] According to some aspects, a circuit quantum electrodynamics
system is provided, comprising nonlinear quantum system comprising
Josephson junction comprising a first superconducting portion, a
second superconducting portion, and an insulating portion, wherein
the first superconducting portion and the second superconducting
portion are physically separated by the insulating portion, and
first antenna electrically connected to the first superconducting
portion, second antenna electrically connected to the first
superconducting portion, and third antenna electrically connected
to the second superconducting portion, a first quantum mechanical
oscillator dispersively coupled to the nonlinear quantum system via
the first antenna, a second quantum mechanical oscillator
dispersively coupled to the nonlinear quantum system via the second
antenna, and at least one electromagnetic radiation source
configured to independently apply electromagnetic pulses to the
nonlinear quantum system, to the first quantum mechanical
oscillator, and to the second quantum mechanical oscillator.
[0131] According to some embodiments, the first quantum mechanical
oscillator comprises a first microwave resonator, and the second
quantum mechanical oscillator comprises a second microwave
resonator,
[0132] According to some embodiments, the first microwave resonator
is a first three-dimensional superconducting cavity, and the second
microwave resonator is a second three-dimensional superconducting
cavity.
[0133] According to some embodiments, the circuit quantum
electrodynamics system further comprises a readout resonator
capacitively coupled to the third antenna of the nonlinear quantum
system.
[0134] According to some embodiments, the nonlinear quantum system
is disposed on a first chip and at least a portion of the readout
resonator is formed on the first chip.
[0135] According to some aspects, a nonlinear quantum device is
provided comprising a Josephson junction comprising a first
superconducting portion, a second superconducting portion, and an
insulating portion, wherein the first superconducting portion and
the second superconducting portion are physically separated by the
insulating portion, and a first antenna electrically connected to
the first superconducting portion, a second antenna electrically
connected to the first superconducting portion, and a third antenna
electrically connected to the second superconducting portion.
[0136] According to some embodiments, the first antenna, the second
antenna and the first superconducting portion intersect at a single
location.
[0137] According to some embodiments, the nonlinear quantum device
further comprises a metallic strip capacitively coupled to the
third antenna.
[0138] Various aspects of the present invention may be used alone,
in combination, or in a variety of arrangements not specifically
discussed in the embodiments described in the foregoing and is
therefore not limited in its application to the details and
arrangement of components set forth in the foregoing description or
illustrated in the drawings. For example, aspects described in one
embodiment may be combined in any manner with aspects described in
other embodiments.
[0139] Also, the invention may be embodied as a method, of which an
example has been provided. The acts performed as part of the method
may be ordered in any suitable way. Accordingly, embodiments may be
constructed in which acts are performed in an order different than
illustrated, which may include performing some acts simultaneously,
even though shown as sequential acts in illustrative
embodiments.
[0140] Various inventive concepts may be embodied as at least one
non-transitory computer readable storage medium (e.g., a computer
memory, one or more floppy discs, compact discs, optical discs,
magnetic tapes, flash memories, circuit configurations in Field
Programmable Gate Arrays or other semiconductor devices, etc.) or a
computer readable storage device encoded with one or more programs
that, when executed on one or more computers or other processors,
implement some of the various embodiments of the present invention.
The non-transitory computer-readable medium or media may be
transportable, such that the program or programs stored thereon may
be loaded onto any computer resource to implement various aspects
of the present invention as discussed above.
[0141] The terms "program," "software," and/or "application" are
used herein in a generic sense to refer to any type of computer
code or set of computer-executable instructions that can be
employed to program a computer or other processor to implement
various aspects of embodiments as discussed above. Additionally, it
should be appreciated that according to one aspect, one or more
computer programs that when executed perform methods of one or more
embodiments described herein need not reside on a single computer
or processor, but may be distributed in a modular fashion among
different computers or processors to implement various aspects of
the present invention.
[0142] Use of ordinal terms such as "first," "second," "third,"
etc., in the claims to modify a claim element does not by itself
connote any priority, precedence, or order of one claim element
over another or the temporal order in which acts of a method are
performed, but are used merely as labels to distinguish one claim
element having a certain name from another element having a same
name (but for use of the ordinal term) to distinguish the claim
elements.
[0143] Also, the phraseology and terminology used herein is for the
purpose of description and should not be regarded as limiting. The
use of "including," "comprising," or "having," "containing,"
"involving," and variations thereof herein, is meant to encompass
the items listed thereafter and equivalents thereof as well as
additional items.
* * * * *